Ranks of rational points of the Jacobian varieties of hyperelliptic curves

2017/02/25 by Im, Bo-Hae, Kim, Byoung Du
#11G10 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R23

paper · doi:10.48550/arxiv.1702.07837

Abstract

In this paper, we obtain bounds for the Mordell-Weil ranks over cyclotomic extensions of a wide range of abelian varieties defined over a number field F whose primes above p are totally ramified over F/ℚ. We assume that the abelian varieties may have good non-ordinary reduction at those primes. Our work is a generalization of \citeKim, in which the second author generalized Perrin-Riou's Iwasawa theory for elliptic curves over ℚ with supersingular reduction (\citePerrin-Riou) to elliptic curves defined over the above-mentioned number field F. On top of non-ordinary reduction and the ramification of the field F, we deal with the additional difficulty that the dimensions of the abelian varieties can be any number bigger than 1 which causes a variety of issues. As a result, we obtain bounds for the ranks over cyclotomic extensions ℚ(μ_p^max(M,N)+n) of the Jacobian varieties of \it ramified hyperelliptic curves y^2pM=x^3pN+ax^pN+b among others.

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